T. Coulhon, G. Kerkyacharian & P. Petrushev

Transcription

1 Heat Kernel Generated Frames in the Setting of Dirichlet Spaces T. Coulhon, G. Kerkyacharian & P. Petrushev Journal of Fourier Analysis and Applications ISSN DOI /s

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3 DOI /s Heat Kernel Generated Frames in the Setting of Dirichlet Spaces T. Coulhon G. Kerkyacharian P. Petrushev Received: 30 June 2011 / Revised: 11 ay 2012 Springer Science+Business edia, LLC 2012 Abstract Wavelet bases and frames consisting of band limited functions of nearly exponential localization on R d are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and utilization, and providing sparse representation of natural function spaces e.g. Besov spaces) on R d. Such frames are also available on the sphere and in more general homogeneous spaces, on the interval and ball. The purpose of this article is to develop band limited well-localized frames in the general setting of Dirichlet spaces with doubling measure and a local scale-invariant Poincaré inequality which lead to heat kernels with small time Gaussian bounds and Hölder continuity. As an application of this construction, band limited frames are developed in the context of Lie groups or homogeneous spaces with polynomial volume growth, complete Riemannian manifolds with Ricci curvature bounded from below and satisfying the volume doubling property, and other settings. The new frames are used for decomposition of Besov spaces in this general setting. Keywords Heat kernel Gaussian bounds Functional calculus Sampling Frames Besov spaces Communicated by Hans G. Feichtinger. P. Petrushev has been supported by NSF Grant DS T. Coulhon Equipe AG, CNRS-UR 8088, Université de Cergy-Pontoise, Cergy-Pontoise, France thierry.coulhon@u-cergy.fr G. Kerkyacharian Laboratoire de Probabilités et odèles Aléatoires, CNRS-UR 7599, Université Paris VI et Université Paris VII, rue de Clisson, Paris, France kerk@math.jussieu.fr P. Petrushev ) Department of athematics, University of South Carolina, Columbia, SC 29208, USA pencho@math.sc.edu

4 athematics Subject Classification 58J35 42C15 43A85 46E35 1 Introduction Decomposition systems bases or frames) consisting of band limited functions of nearly exponential space localization have had significant impact in theoretical and computational harmonic analysis, PDEs, statistics, approximation theory and their applications. eyer s wavelets [39] and the frames the ϕ-transform) of Frazier and Jawerth [21 23] are the most striking examples of such decomposition systems playing a pivotal role in the solution of numerous theoretical and computational problems. The key to the success of wavelet type bases and frames is rooted in their ability to capture a great deal of smoothness and other norms in terms of respective coefficient sequence norms and provide sparse representation of natural function spaces e.g. Besov spaces) on R d. Frames of a similar nature have been recently developed in non-standard settings such as on the sphere [42, 43] and more general homogeneous spaces [24], on the interval [35, 47] and ball [36, 48] with weights, and extensively used in statistical applications see e.g. [31, 32]). The primary goal of this paper is to extend and refine the construction of band limited frames with elements of nearly exponential space localization to the general setting of strictly local regular Dirichlet spaces with doubling measure and local scale-invariant Poincaré inequality which lead to a arkovian heat kernel with small time Gaussian bounds and Hölder continuity. The key point of our approach is to be able to deal with a) different geometries, b) compact and noncompact spaces, and c) spaces with nontrivial weights, and at the same time to allow for the development and frame decomposition of Besov and Triebel-Lizorkin spaces with complete range of indices. This will enable us to cover and shed new light on the existing frames and space decompositions and develop band limited localized frames in the context of Lie groups or homogeneous spaces with polynomial volume growth, complete Riemannian manifold with Ricci curvature bounded from below and satisfying the volume doubling condition, and other new settings. To this end we shall make advances on several fronts: development of functional calculus of positive self-adjoint operators with associated heat kernel in particular, localization of the kernels of related integral operators), development of lower bounds on kernel operators, development of a Shannon sampling theory, Littlewood-Paley analysis, and development of dual frames. As a first application of our frames we shall develop rapidly and characterize the classical Besov spaces Bpq s with positive smoothness and p 1. Classical and nonclassical Besov and Triebel-Lizorkin spaces in the general framework of this paper with full range of indices and their frame and heat kernel decompositions are developed in the follow-up paper [33]. In this preamble we outline the main components and points of this undertaking, including the underlying setting, a general scenario for realization of the setting and examples, and a description of the main results.

5 1.1 The Setting We now describe precisely all the ingredients we need to develop our theory. I. We assume that,ρ,μ) is a metric measure space, which satisfies the conditions: a), ρ) is a locally compact metric space with distance ρ, ) and μ is a positive Radon measure such that the following volume doubling condition is valid 0 <μ Bx,2r) ) 2 d μ Bx,r) ) < for x and r>0. 1.1) Here Bx,r) is the open ball centered at x of radius r and d>0 is a constant that plays the role of a dimension. Note that,ρ,μ) is also a homogeneous space in the sense of Coifman and Weiss [10, 11]. b) The reverse doubling condition is assumed to be valid, that is, there exists a constant β>0 such that μ Bx,2r) ) 2 β μ Bx,r) ) for x and 0 <r diam. 1.2) 3 It will be shown in Sect. 2 that this condition is a consequence of the above doubling condition if is connected. c) The following non-collapsing condition will also be stipulated: There exists a constant c>0 such that inf x μ Bx,1) ) c, x. 1.3) As will be shown in Sect. 2 in the case μ) < the above inequality follows by 1.1). Therefore, it is an additional assumption only when μ) =. Since we consider in this paper inhomogeneous function spaces only, it would be natural to make only purely local assumptions, and in particular to assume doubling only for balls with radius bounded by some constant, which would enlarge considerably our range of examples. This would require however more work and more space. On the other hand, our next assumptions on the heat kernel are local, in the sense that they are required for small time only. Clearly, by assuming global doubling and global heat kernel bounds, one can treat homogeneous spaces as well. II. Our main assumption is that the local geometry of the space,ρ,μ)is related to an essentially self-adjoint positive operator L on L 2, dμ) such that the associated semigroup P t = e tl consists of integral operators with heat) kernel p t x, y) obeying the conditions: d) Small time Gaussian upper bound: p t x, y) C exp{ cρ2 x,y) t } μbx, t))μby, t)) for x,y, 0 <t ) One can see that by combining the results in [9, 45] and [13], this estimate and the doubling condition 1.1) coupled with the fact that e tl is actually a holomorphic

6 semigroup on L 2, dμ), i.e.e zl exists for z C, Rez 0, imply that e zl is an integral operator with kernel p z x, y) satisfying the following estimate: For any z = t + iu,0<t 1, u R, x,y, p z x, y) C exp{ c Re ρ2 x,y) z }. 1.5) μbx, t))μby, t)) e) Hölder continuity: There exists a constant α>0 such that p t x, y) p t x,y ) ρy,y ) ) α exp{ cρ2 x,y) t } C 1.6) t μbx, t))μby, t)) for x,y,y and 0 <t 1, whenever ρy,y ) t. f) arkov property: p t x, y)dμy) 1 for t>0, 1.7) which readily implies, by analytic continuation, p z x, y)dμy) 1 for z = t + iu, t>0. 1.8) Above C,c > 0 are structural constants that will affect almost all constants in what follows. The main results in this article will be derived from the above conditions. However, it is perhaps suitable to exhibit a more tangible general scenario that guarantees the validity of these conditions. 1.2 Realization of the Setting in the Framework of Dirichlet Spaces We would like to point out that in a general framework of Dirichlet spaces the needed Gaussian bound, Hölder continuity, and arkov property of the heat kernel follow from the local scale-invariant Poincaré inequality and the doubling condition on the measure, which in turn are equivalent to the parabolic Harnack inequality. The point is that situations where our theory is applicable are quite common and it just amounts to verifying the local scale-invariant Poincaré inequality and the doubling condition on the measure. We shall further illustrate this point on several examples and, in particular, on the simple example of [ 1, 1] with the heat kernel induced by the Jacobi operator, seemingly not covered in the literature. We shall operate in the framework of strictly local regular Dirichlet spaces see [2, 5, 6, 14, 20, 45, 55 57]). To be specific, we assume that is a locally compact separable metric space equipped with a positive Radon measure μ such that every open and nonempty set has positive measure. Also, we assume that L is a positive symmetric operator on the real) L 2, μ) with domain DL), dense in L 2, μ). We shall denote briefly L p := L p, μ) in what follows. One can associate with L a symmetric non-negative form Ef, g) = Lf, g =Eg, f ), Ef, f ) = Lf, f 0,

7 with domain DE) = DL). We consider on DE) the prehilbertian structure induced by f 2 E = f Ef, f ) which in general is not complete not closed), but closable [14]) in L 2. Denote by E and DE) the closure of E and its domain. It gives rise to a self-adjoint extension L the Friedrichs extension) of L with domain DL) consisting of all f DE) for which there exists u L 2 such that Ef, g) = u, g for all g DE) and Lf = u. Then L is positive and self-adjoint, and DE) = D L) 1/2), Ef, g) = L) 1/2 f,l) 1/2 g. Using the classical spectral theory of positive self-adjoint operators, we can associate with L a self-adjoint strongly continuous contraction semigroup P t = e tl on L 2, μ). Then e tl = 0 e λt de λ, where E λ is the spectral resolution associated with L. oreover this semigroup has a holomorphic extension to the complex half-plane Re z>0. Our next assumption is that P t is a submarkovian semigroup: 0 f 1 and f L 2 imply 0 P t f 1. Then P t can be extended as a contraction operator on L p,1 p, preserving positivity, satisfying P t 1 1, and hence yielding a strongly continuous contraction semigroup on L p, 1 p<. A sufficient condition for this [2, 20], which can be verified on DL), isthatforeveryε>0 there exists Φ ε : R [ ε, 1 + ε] such that Φ ε is non-decreasing, Φ ε Lip 1, Φ ε t) = t for t [0, 1] and Φ ε f ) DE) and E Φ ε f ), Φ ε f ) ) Ef, f ), f DL) in fact, this can be done easily only if Φ ε f ) DL)). Under the above conditions, DE), E) is called a Dirichlet space and DE) L is an algebra. We assume that the form E is strongly local, i.e. Ef, g) = 0forf,g DE) whenever f is with compact support and g is constant on a neighbourhood of the support of f. We also assume that E is regular, meaning that the space C c ) of continuous functions on with compact support has the property that the algebra C c ) DE) is dense in C c ) with respect to the sup norm, and dense in DE) in the norm Ef, f ) + f 2 2. We next give a sufficient condition for strong locality and regularity [20], Chap. 3) which can be verified for DL) : E is strongly local and regular if i) DL) is a subalgebra of C c ) verifying the strong local condition: 0 = Ef, g) = Lf, g if f,g DL), f is with compact support, and g is constant on a neighbourhood of the support of f, and ii) for any compact K and open set U such that K U there exists u DL), u 0, supp u U, and u 1onK thus DL) is a dense subalgebra of C c ) and dense in DE)).

8 Under the above assumptions, there exists a bilinear symmetric form dγ defined on DE) DE) with values in the signed Radon measures on such that Eφf, g) + Ef, φg) Eφ, fg) = 2 φdγ f,g) for f,g,φ C c ) DE), which obviously verifies Ef, g) = dγf,g) and dγf,f) 0. In fact, if DL) is a subalgebra of C c ), then dγ is absolutely continuous with respect to μ, and dγ f, g)u) = Γ f, g)u)dμu), Γf,g)= 1 ) Lfg) flg glf 2 f,g DL). In other words, E admits a carré du champ [8], Chap. 1, Sect. 4): There exists a bilinear function DE) DE) f,g Γf,g) L 1 such that Γ f, f )u) 0, Eφf, g) + Ef, φg) Eφ, fg) = 2 φu)γ f, g)u)dμu) f,g,φ DE) L, and Ef, g) = 2 Γ f, g)u)dμu). One can define an intrinsic distance on by ρx,y) = sup { ux) uy) : u DE) C c ), dγ u, u) = γ u)x)dμx), γ u)x) 1 }. We assume that ρ : [0, ] is actually a true metric that generates the original topology on and that, ρ) is a complete metric space. As a consequence of this assumption, the space is connected, the closure of an open ball Bx,r) is the closed ball Bx,r) := {y,ρx,y) r}, and the closed balls are compact see [55 57]). We are now in a position to describe an optimal scenario when the needed Gaussian bound 1.4), Hölder continuity 1.5), and arkov property 1.6) on the heat kernel can be effectively realized. In the framework of strictly local regular Dirichlet spaces with a complete intrinsic metric, the following two properties are equivalent [28, 57]: i) The heat kernel satisfies c 1 exp{ c 1ρ 2 x,y) t } c 2 p t x, y) exp{ c 2ρ 2 x,y) t } 1.9) μbx, t))μby, t)) μbx, t))μby, t)) for x,y and 0 <t 1. ii)a),ρ,μ) is a local doubling measure space: There exists d>0 such that μbx, 2r)) 2 d μbx, r)) for x and 0 <r<1. b) Local scale-invariant Poincaré inequality holds: There exists a constant C>0 such that for any ball B = Bx,r) with 0 <r 1, x, and any function

9 f DE), B f f B 2 Cr 2 B dγ f, f ). Here f B is the mean of f over B. oreover, it is also well-known that the above property is equivalent to a local parabolic Harnack inequality, and, furthermore, any of these equivalent properties implies the validity of 1.6) and 1.7)see[27, 28, 53, 57], and the references therein). Consequently, given a situation which fits into the framework of strictly local regular Dirichlet spaces with a complete intrinsic metric it suffices to only verify the local Poincaré inequality and the global doubling condition on the measure and then our theory applies in full. In a future work we shall further develop this theory under the more general assumption of the small time sub-gaussian estimate: C exp{ c ρm x,y) m 1 } p t x, y) μbx, t 1/m ))μby, t 1/m )) t ) 1 for x,y, 0 <t 1, 1.10) where m Examples There is a great deal of set-ups which fit in the general framework of this article. We next briefly describe several benchmark examples which are indicative for the versatility and depth of our methods Uniformly Elliptic Divergence form Operators on R d Given a uniformly elliptic symmetric matrix-valued function {a i,j x)} depending on x R d, one can define an operator L = d i,j=1 x i a i,j ) x j on L 2 R d,dx) via the associated quadratic form. Thanks to the uniform ellipticity condition, the intrinsic metric associated with this operator is equivalent to the Euclidean distance. The Gaussian upper and lower estimates of the heat kernel in this setting hold for all time and are due to Aronson, the Hölder regularity of the solutions is due to Nash [44], the Harnack inequality was obtained by oser [40, 41] Domains in R d One can define uniformly elliptic divergence form operators on R d by choosing boundary conditions. In this case the upper bounds of the heat kernels are well understood see for instance [45]). The problem for establishing Gaussian lower bounds is much more complicated. One has to choose Neumann conditions and impose regularity assumptions on the domain. For the state of the art, we refer the reader to [27].

10 1.3.3 Riemannian anifolds and Lie Groups The conditions from Sect. 1.2 are verified for the Laplace-Beltrami operator of a Riemannian manifold with non-negative Ricci curvature [38], also for manifolds with Ricci curvature bounded from below if one assumes in addition that they satisfy the volume doubling property, also for manifolds that are quasi-isometric to such a manifold [25, 51, 52], also for co-compact covering manifolds whose deck transformation group has polynomial growth [51, 52], for sublaplacians on polynomial growth Lie groups [50, 61] and their homogeneous spaces [39]. We would like to point out that the case of the sphere endowed with the natural Laplace-Beltrami operator treated in [42, 43] and the case of more general compact homogeneous spaces endowed with the Casimir operator considered in [24] fall into the above category. One can also consider variable coefficients operators on Lie groups, see [54]. We refer the reader to [27, Sect. 2.1] for further details on the above examples. For more references on the heat kernel in various settings, see [14, 26, 53, 61] Heat Kernel on [ 1, 1] Generated by the Jacobi Operator To show the flexibility of our general approach to frames and spaces through heat kernels we consider in Sect. 7 the simple example of =[ 1, 1] with dμx) = w α,β x)dx, where w α,β x) is the classical Jacobi weight: w α,β x) = wx) = 1 x) α 1 + x) β, α,β > 1. The Jacobi operator is defined by Lf x) = [wx)ax)f x)] wx) with ax) := 1 x 2 and DL) = C 2 [ 1, 1]. As is well-known [58], LP k = λ k P k, where P k k 0) is the kth degree normalized) Jacobi polynomial and λ k = kk + α + β + 1). Integration by parts gives 1 Ef, g) := Lf, g = ax)f x)g x)w α,β x)dx. 1 In Sect. 7 it will be shown that in this case the general theory applies, resulting in a complete strictly local Dirichlet space with an intrinsic metric defined by ρx,y) = arccos x arccos y, x,y [ 1, 1], which is apparently compatible with the usual topology on [ 1, 1]. It will be also shown that in this setting the measure μ verifies the doubling condition and the respective scale-invariant Poincaré inequality is valid. Therefore, the example under consideration fits in the general setting described in Sect. 1.2 and our theory applies. In particular, the associated heat kernel with a representation p t x, y) = k 0 e λ kt P k x)p k y)

11 has Gaussian bounds see Sect. 7), which to the best of our knowledge appears first in the present article. Another consequence of this is that our theory covers completely the construction of frames and the development of Besov and Triebel-Lizorkin spaces on [ 1, 1] with Jacobi weights from [35, 47]. Finally, we would like to point out that there are other examples, e.g. the development of frames and weighted Besov and Triebel-Lizorkin spaces on the unit ball B in R d in [36, 48], which perfectly fit in our general setting, but we shall not pursue in this article. 1.4 Outline of the Paper This paper is organized as follows: In Sect. 2 we give some auxiliary results which are instrumental in proving our main results. In particular, we collect all needed facts about doubling measures and related kernels, construction of maximal δ-nets, and integral operators. In Sect. 3 we develop some components of a non-holomorphic functional calculus related to a positive self-adjoint operator L in the general set-up of the paper. In particular, we establish the nearly exponential localization of the kernels of operators of the form f L) under suitable conditions on f. These localization results are crucial for the development of the Littlewood-Paley theory in our setting. They also enable us to explore the main properties of the spectral spaces and develop the linear approximation theory from spectral spaces through the machinery of Jackson-Bernstein inequalities and interpolation. In this section we also give the main properties of finite dimensional spectral spaces. In Sect. 4 we establish a sampling theorem in the spirit of the Shannon theory and develop a cubature rule/formula in the compact and non-compact case, which is exact for spectral functions of a given order. This cubature rule is a critical component in the development of our frames. Our main results are placed in Sect. 5, where we construct pairs of dual frames of the form: {ψ jξ : ξ X j,j 0}, { ψ jξ : ξ X j,j 0}, where each X j is a δ j -net on for an appropriate δ j. The frame elements ψ jξ, ψ jξ are band limited and well-localized functions, which allow for decomposition of functions and distributions from various spaces in particular, Besov and Triebel-Lizorkin spaces) of the form f = f, ψ jξ ψ jξ. j 0 ξ X j The most critical point in this paper is the construction of the dual frame { ψ jξ }. We develop it in two settings: i) in the general case, and ii) in the case when the spectral spaces have the polynomial property under multiplication see Sect. 5.3). In the second case the construction is simple and elegant, however, the setting is somewhat restrictive, while in the first case the construction is much more involved, but the localization of ψ jξ is inverse polynomial of an arbitrarily fixed order.

12 In Sect. 6 we develop the classical and most commonly used Besov spaces Bpq s with indices s>0, 1 p, and 0 <q in the setting of this paper. These spaces are defined through Littlewood-Paley decomposition and characterized as approximation spaces of linear approximation from spectral spaces. A frame decomposition of Bpq s is also established. In full generality, classical and non-classical Besov and Triebel-Lizorkin spaces and their frame decomposition in the general setting of the paper are developed in [33]. Section 7 is an Appendix, where we place the proofs of the Poincaré inequality for the Jacobi operator and the doubling property of the respective measure. Gaussian bounds of the associated heat kernel are also established. Notation Throughout this article we shall use the notation E :=μe) for E, L p := L p, μ), p := L p, and T p q will denote the norm of a bounded operator T : L p L q. UCB will stand for the space of all uniformly continuous and bounded functions on and L will be in most cases identified with UCB. DT ) will stand for the domain of a given operator T. We shall denote by C 0 R +) the set of all compactly supported C functions on R + := [0, ). In most cases sup will mean ess sup. Positive constants will be denoted by c, C, c 1, c,... and they may vary at every occurrence, a b will stand for c 1 a/b c 2. 2 Doubling etric easure Spaces: Basic Facts In this section we put together some simple facts related to metric measure spaces,ρ,μ) obeying the doubling, inverse doubling and non-collapsing conditions 1.1) 1.3) and integral operators acting on functions defined on such spaces. 2.1 Consequences of Doubling and Clarifications The doubling condition 1.1) readily implies Bx,λr) 2λ) d Bx,r), x, λ > 1, r>0, 2.1) and, therefore, due to Bx,r) By,ρy,x) + r) Bx,r) 2 d 1 + ρx,y) ) d By,r), r x,y, r > ) In turn, the reverse doubling condition yields Bx,λr) λ/2) β Bx,r), λ>1, r>0, 0 <λr< diam. 2.3) 3 Also, the non-collapsing condition 1.3) coupled with 2.1) implies inf Bx,r) ĉr d, 0 <r 1, 2.4) x where ĉ = c2 d with c>0 the constant from 1.3).

13 Note that Bx,r) can be much larger than cr d as is evidenced by the case of the Jacobi operator on [ 1, 1], considered in Sects and 7, see7.1). Several clarifying statements are in order. We begin with a claim which, in particular, shows that the non-collapsing condition is automatically obeyed when μ) <. Proposition 2.1 Let,ρ,μ) be a metric measure space which obeys the doubling condition 1.1). Then a) μ) < if and only if diam <. oreover, if diam = D<, then inf Bx,r) r d 2D) d, 0 <r D. 2.5) x b) μ{x})>0 for some x if and only if {x}=bx,r) for some r>0. Proof We first prove a). Note that if diam = D<, then = Bx,D) for any x and hence = Bx,D) <. In the other direction, let <. Assume on the contrary that diam =. Then inductively one can construct a sequence of points {x 0,x 1,...} such that if d j := ρx 0,x j ), then 1 d 1 <d 2 < and d j+1 > 3d j, j 0. One checks easily that Bx j, d j 2 ) Bx k, d k 2 ) = if j k. On the other hand, using 1.1), 0 < Bx0, 1) Bxj, 2d j ) 4 d Bxj,d j /2). Therefore, we have a sequence of disjoint balls {Bx j, d j 2 )} j 1 in such that Bx j, d j 2 ) 4 d Bx 0, 1) > 0 and hence =. This is a contradiction that proves the claim. Estimate 2.5) is immediate from 2.1). To prove b), we first note that if {x}=bx,r) for some r>0, then 1.1) implies μ{x})>0. For the other implication, let μ{x})>0 and assume that {x} Bx,r) for all r>0. Then we use this to construct inductively a sequence {x 1,x 2,...} such that if d j := ρx,x j ), then d 1 >d 2 > > 0 and d j+1 < d j 3, j 1. Clearly, the latter inequality yields Bx j, d j 2 ) Bx k, d k 2 ) = if j k. On the other hand by our assumption, 1.1), and the fact that x Bx j, 2d j ) we infer 0 <μ {x} ) Bx j, 2d j ) 4 d Bx j,d j /2). Now, as above we conclude that = which is a contradiction. We next show that the reverse doubling condition 1.2) is not quite restrictive. Proposition 2.2 If is connected, then the reverse doubling condition holds, i.e. there exists β>0 such that Bx,2r) 2 β Bx,r) for x and 0 <r< diam 3.

14 Proof Suppose 0 <r< diam 3. Then there exists y such that dx,y) = 3r/2, for otherwise Bx,3r/2) = Bx,3r/2) is simultaneously open and close, which contradicts the connectedness of. Evidently, Bx,r) By,r/2) = and By,r/2) Bx,2r), which yields Bx,2r) By,r/2) + Bx,r). On the other hand Bx,r) By,5r/2) which along with 2.1) implies Bx,r) 10 d By,r/2) and hence Bx,2r) 10 d + 1) Bx,r) =2 β Bx,r). 2.2 Useful Notation and Estimates The localization of various operator kernels in what follows will be governed by symmetric functions of the form D δ,σ x, y) := Bx,δ) By,δ) ) 1/2 1 + ρx,y) ) σ, x,y. 2.6) δ Here δ,σ > 0 are parameters that will be specified in every particular case. We next give several simple properties of D δ,σ x, y) which will be instrumental in various proofs in the sequel. Note first that 2.1) 2.2) readily yield D δ,σ x, y) 2 d/2 Bx,δ) ρx,y) ) σ d/2, 2.7) δ D λδ,σ x, y) 2/λ) d D δ,σ x, y), 0 <λ<1, 2.8) D λδ,σ x, y) λ σ D δ,σ x, y), λ > ) Furthermore, for 0 <p< and σ>d1/2 + 1/p) D δ,σ x, ) [ p = Dδ,σ x, y) ] ) 1/p p dμy) 2 dp/2 cp) Bx,δ) 1/p 1, 2.10) where cp) = ) 1/p is decreasing as a function of p, and 2 d 2 σ d/2)p D δ,σ x, u)d δ,σ u, y)dμu) cd δ,σ x, y) if σ>2d, 2.11) 2σ +d+1 with c =. 2 d 2 d σ The above two estimates follow readily by the following lemma which will be needed as well. Lemma 2.3 a) If σ>d, then for δ>0 1 + δ 1 ρx,y) ) σ dμy) c1 Bx,δ), x c 1 = 2 d 2 σ ) 1). 2.12)

15 b) If σ>d, then for x,y and δ> δ 1 ρx,u)) σ 1 + δ 1 ρy,u)) σ dμu) 2 σ c 1 Bx,δ) + By,δ) 1 + δ 1 ρx,y)) σ 2 σ 2 d + 1 ) Bx,δ) c δ ) ρx,y)) σ d c) If σ>2d, then for x,y and δ>0 1 Bu,δ) 1 + δ 1 ρx,u)) σ 1 + δ 1 ρy,u)) σ dμy) with c 2 = 2σ +d+1. 2 d 2 d σ c δ 1 ρx,y)) σ, 2.14) Proof Denote briefly E 0 := {y : ρx,y) < δ}=bx,δ) and E j := { y : 2 j 1 δ ρx,y) < 2 j δ } = B x,2 j δ ) \ B x,2 j 1 δ ), j 1. Then using 1.1) we get 1 + δ 1 ρx,y) ) σ dμy) = 1 + δ 1 ρx,y) ) σ dμy) j 0 E j Bx,δ) + 2 d 1 ) j 0 Bx,δ) d 1 ) Bx,2 j δ) j ) σ j 0 2 jd ) j ) σ Bx,δ) 2 d 2 σ, which gives 2.12). For the proof of 2.13), we note that the triangle inequality implies 1 + δ 1 ρx,y) 1 + δ 1 ρx, u))1 + δ 1 ρy,u)) δ 1 ρx,u) δ 1 ρy,u)

16 and hence 1 + δ 1 ρx,y)) σ 1 + δ 1 ρx,u)) σ 1 + δ 1 ρy,u)) σ 2 σ 1 + δ 1 ρx,u)) σ + 2 σ 1 + δ 1 ρy,u)) σ. 2.15) We now integrate and use 2.12) to obtain 2.13). For the proof of 2.14), we use the above inequality and 2.2) to obtain 1 + δ 1 ρx,y)) σ Bu,δ) 1 + δ 1 ρx,u)) σ 1 + δ 1 ρy,u)) σ 2 σ +d Bx,δ) 1 + δ 1 ρx,u)) σ d + 2 σ +d By,δ) 1 + δ 1 ρy,u)) σ d 2.16) and integrating and applying again 2.12) we arrive at 2.14). 2.3 aximal δ-nets For the construction of decomposition systems frames) we shall need maximal δ-nets on. Definition 2.4 We say that X is a δ-net on δ>0) if ρξ,η) δ for all ξ,η X, and X is a maximal δ-net on if X is a δ-net on that cannot be enlarged, i.e. there does not exist x such that ρx,ξ) δ for all ξ X and x X. We collect some simple properties of maximal δ-nets in the following proposition. Proposition 2.5 Suppose,ρ,μ)is a metric measure space obeying the doubling condition 1.1) and let δ>0. a) A maximal δ-net on always exists. b) If X is a maximal δ-net on, then = Bξ,δ) and Bξ,δ/2) Bη,δ/2) = if ξ η, ξ,η X. 2.17) ξ X c) Let X be a maximal δ-net on. Then X is countable or finite and there exists a disjoint partition {A ξ } ξ X of consisting of measurable sets such that Bξ,δ/2) A ξ Bξ,δ), ξ X. 2.18) Proof For a) observe that a maximal δ-net is a maximal set in the collection of all δ-net on with respect to the natural ordering of sets by inclusion) and hence by Zorn s lemma a maximal δ-net on exists. Part b) is immediate from the definition of maximal δ-nets.

17 To prove c) we first fix y and observe that for any n>δ, n N, by2.1) 2.2) it follows that By,n) cn,δ) Bξ,δ/2) for ξ X By,n), where cn,δ) is a constant depending on n and δ. On the other hand, by 2.17) ξ X By,n) Bξ,δ/2) By,2n) 2 d By,n). Therefore, #X By,n)) 2 d cn,δ) <, which readily implies that X is countable or finite. Let us order the elements of X in a sequence: X ={ξ 1,ξ 2,...}. We now define the sets A ξ of the claimed cover of inductively. We set A ξ1 := Bξ 1,δ)\ η X,η ξ 1 Bη,δ/2) and if A ξ1,a ξ2,...,a ξj 1 have already been defined, we set A ξj := Bξ j,δ) [ ] A ξν Bη,δ/2). ν j 1 η X,η ξ j It is easy to see that the sets A ξ1,a ξ2,... have the claimed properties. Discrete versions of estimates 2.11) and 2.12) will be needed. Suppose X is a maximal δ-net on and {A ξ } ξ X is a companion disjoint partition of as in Proposition 2.5. Then A ξ 1 + δ 1 ρx,ξ) ) d 1 2 2d+2 Bx,δ) 2.19) ξ X and 1 + δ 1 ρx,ξ) ) 2d 1 2 3d ) ξ X Furthermore, for any δ δ A ξ δ ρx,ξ) ) 2d 1 2 3d+2, 2.21) Bξ,δ ) ξ X and if σ 2d + 1 A ξ D δ,σ x, ξ)d δ,σ y, ξ) 2 σ +3d+3 D δ,σ x, y). 2.22) ξ X Also, for σ 2d δ 1 ρx,ξ) ) σ 1 + δ 1 ρy,ξ) ) σ 2 σ +2d δ 1 ρx,y) ) σ. 2.23) ξ X

18 We next prove 2.21). The proofs of 2.19) and 2.20) are similar. Observe first that by 2.2) Bx,δ ) 2 d 1 + δ 1 ρx,ξ)) d Bξ,δ ). On the other hand, for u A ξ Bξ,δ) 1 + δ 1 ρx,u) 1 + δ 1 ρx,ξ)+ δ 1 ρξ,u) δ 1 ρx,ξ) ). Therefore, A ξ δ ρx,ξ) ) 2d 1 Bξ,δ ) 2d A ξ δ ρx,ξ) ) d 1 Bx,δ ) 22d δ ρx,u) ) d 1 dμu). Bx,δ ) A ξ This leads to A ξ δ ρx,ξ) ) 2d 1 Bξ,δ ) ξ X j 22d δ ρx,u) ) d 1 dμu) 2 3d+2, Bx,δ ) where for the last inequality we used 2.12). Thus 2.21) is established. For the proof of 2.22), we observe that using 2.15) A ξ D δ,σ x, ξ)d δ,σ y, ξ) A ξ 1 + δ 1 ρx,y)) σ = D δ,σ x, y) Bξ,δ) 1 + δ 1 ρx,ξ)) σ 1 + δ 1 ρy,ξ)) σ [ D δ,σ x, y) 2 σ A ξ Bξ,δ ) 1 + δ 1 ρx,ξ)) σ + 2 σ A ξ Bξ,δ ) 1 + δ 1 ρy,ξ)) σ Now, summing up and applying 2.21) we arrive at 2.22). Estimate 2.23) follows in a similar manner from 2.15) and 2.20). 2.4 Integral Operators We shall mainly deal with integral kernel) operators. The kernels of many operators will be controlled by the quantities D δ,σ x, y), introduced in 2.6). Our first order of business is to establish a Young-type inequality for such operators. Proposition 2.6 Let H be an integral operator with kernel Hx,y), i.e. Hf x) = Hx,y)fy)dμy), and let Hx,y) c D δ,σ x, y) ].

19 for some 0 <δ 1 and σ 2d + 1. If 1 p q, then Hf q cδ d 1 q 1 p ) f p, f L p, 2.24) where c = c ĉ d1/r 1) 2 2d+1 with ĉ being the constant from 2.4). This result is immediate from the following well-known lemma. Lemma 2.7 Suppose p 1 q 1 = 1 1 r,1 p,q,r, and let Hx,y) be a measurable kernel, verifying the conditions H,y) r K and Hx, ) r K. 2.25) If Hf x) = Hx,y)fy)dμy), then Hf q K f p for f L p. For the proof, see e.g. [19, Theorem 6.36]. Proof of Proposition 2.6 Pick 1 r so that 1/p 1/q = 1 1/r. By2.10) and 2.4) we obtain H,y) r c cr) By,δ) 1/r 1 c c1) ĉδ) d1/r 1) and a similar estimate holds for Hx, ) r. These estimates and the above lemma imply 2.24). We shall frequently use the following well-known result [16], Theorem 6, p. 503). Proposition 2.8 An operator T : L 1 L is bounded if and only if T is an integral operator with kernel K L ), i.e. Tfx) = Kx,y)fy)dμy) a.e. on, and if this is the case T 1 = K L. oreover, the boundedness of T can be expressed in the bilinear form Tf,g c f L 1 g L 1, f,g L 1. We next use this to derive a useful result for products of integral and non-integral operators. Proposition 2.9 In the general setting of a doubling metric measure space,ρ,μ), let U,V : L 2 L 2 be integral operators and suppose that for some 0 <δ 1 and σ d + 1 we have Ux,y) c 1 D δ,σ x, y) and Vx,y) c 2 D δ,σ x, y). 2.26)

20 Let R : L 2 L 2 be a bounded operator, not necessarily an integral operator. Then URV is an integral operator with the following upper bound on its kernel URVx,y) Ux, ) 2 R 2 2 V,y) c R ) Bx,δ) By,δ) with c := c 1 c 2 2 2d+1. Proof By Proposition 2.6 we get URV 1 U 2 R 2 2 V 1 2 cδ d R 2 2 and, therefore, URV is a kernel operator. Formally, we have URV )f = Ux, u)rv )f u)dμu) = Ux,u) R [ V,y) ] u)f y)dμy)dμu) = Ux,u)R [ V,y) ] ) u)dμu) fy)dμy) 2.28) and hence the kernel of URV is given by Hx,y) = Ux,u)R [ V,y) ] u)dμu) = Ux, ), R [ V,y) ]. 2.29) This along with 2.26) and 2.10) leads to Hx,y) Ux, ) 2 R [ V,y) ] 2 c 1c 2 [c2)] 2 R 2 2 Bx,δ) 1/2 By,δ) 1/2, which confirms 2.27), taking into account that [c2)] 2 2 2d+1 by 2.10) ifσ d + 1. It remains to justify the manipulations in 2.28). Observe first that in order to prove 2.29) it suffices to establish identities 2.28) for all f L 2 such that supp f Ba,R) an arbitrary ball on. To this end we shall need Bochner s integral. In particular, we shall use the following results e.g. [62], pp ): Suppose B is a separable Banach space and F :,μ,σ) B is measurable in the following sense: l B, x lf x)) is measurable. Then Bochner s integral B) Fx)dμx) is well defined and takes its value in B if and only if Fx) B dμx) <. Furthermore, if B) Fx)dμx)exists, then l B) F x)dμx)) = lf x))dμx) for any l B. Also, if T : B B is a bounded linear operator, then B) ) B) T Fx)dμx) = T Fx) ) dμx). 2.30)

21 We shall utilize Bochner s integral in our setting with B = L 2. Suppose f L 2 and supp f Ba,R), a, R>0. Then using 2.26), 2.10), and 2.2) we obtain V, y)f y) 2 dμy) c fy) By,δ) 1/2 dμy) Ba,R) c f 2 Ba,R) c f 2 Ba,δ) By,δ) 1 dμy) Ba,R) ) 1/2 1 + δ 1 ρy,a) ) d dμy) ) 1/2 <. 2.31) Therefore, B) V, y)f y)dμy) exists and for any g L2 B) V, y)f y)dμy), g = = = Vf,g. gx)vx,y)dμx) fy)dμy) ) gx) Vx,y)fy)dμy) dμx)) Here the shift of the order of integration is justified by Fubini s theorem and the fact that Vx,y) fy) gx) dμx)dμy) g 2 V,y) fy) dμy) g 2 2 ) V, y)f y) 2 dμy) <, whereweused2.31). Therefore, Vf = B) V, y)f y)dμy). Wenowuse2.30) to obtain [ B) ] B) RVf = R V, y)f y)dμy) = R [ V,y) ] f y)dμy), which implies URV )f x) = B) Ux, u)rv )f u)dμu) R [ V,y) ] f y)dμy), Ux, ) = = Ux,u)R [ V,y) ] ) u)dμu) f y)dμy). Consequently, Hx,y) is given by 2.29) and the proof is complete.

22 3 Functional Calculus The aim of this section is to develop the functional calculus of operators of the form f L) associated with smooth and non-smooth functions f. The calculus of smooth operators is in the spirit of [17, 45] and will be needed in most part of this article, including the construction of frames and the Littlewood-Paley theory, while the nonsmooth calculus will be needed for estimation of the kernels of the spectral projectors and lower bound estimates. 3.1 Smooth Functional Calculus We shall be operating in the setting described in Sect ore precisely, we assume that,ρ,μ) is a metric measure space obeying conditions 1.1) 1.3) and L is an essentially self-adjoint positive operator on L 2 such that the semi-group e tl, t>0, has a kernel p t x, y) verifying 1.4) 1.8). Theorem 3.1 Let g : R C be a measurable function such that for some σ>2d g := ĝξ) 1 + ξ ) σ dξ <, where ĝξ) := gx)e ixξ dx 3.1) R is the Fourier transform of g. Then gδ 2 L)e δ2l,0<δ 1, is an integral operator with kernel gδ 2 L)e δ2l x, y) satisfying g δ 2 L ) e δ2l x, y) c σ g D δ,σ x, y), x,y, 3.2) and g δ 2 L ) e δ2l x, y) g δ 2 L ) e δ2 L x,y ) ρy,y ) ) α c σ g D δ,σx, y), δ 3.3) for all x,y,y, if ρy,y ) δ. Here α>0is the constant from 1.6), D δ,σ x, y) is defined in 2.6), and c σ > 0 is a constant depending only on σ and the structural constants from 1.5) 1.6). oreover, g δ 2 L ) e δ2l x, y)dμy) = g0) x. 3.4) Proof To prove 3.2) we first show that gδ 2 L)e δ2l is a kernel operator. From 3.1) it follows that ĝ 1 < which implies gx) = 2π 1 R ĝξ)eixξ dx and hence g 1 2π ĝ 1. Then by the spectral theorem g δ 2 L ) e δ2 L 2 2 = g δ 2 )e δ2 2π) 1 ĝ 1. Therefore, invoking Proposition 2.8, in order to show that gδ 2 L)e δ2l is a kernel operator it suffices to prove that g δ 2 L ) e δ2l ϕ,ψ c ϕ 1 ψ 1, ϕ,ψ L 1 L 2. R

23 Let E λ, λ 0, be the spectral resolution associated with the operator L, then L = 0 λde λ. Writing the spectral decomposition of gδ 2 L)e δ2l and using the Fourier inversion identity, we obtain for ϕ,ψ L 1 L 2 g δ 2 L ) e δ2l ϕ,ψ = g δ 2 λ ) e δ2λ d E λ ϕ,ψ 0 1 = ĝξ)e iδ2λξ dξ 0 2π R = 1 ĝξ) 2π = 1 2π R R 0 ) e δ2λ d E λ ϕ,ψ ) e δ2λ1 iξ) d E λ ϕ,ψ dξ ĝξ) e δ2 1 iξ)l ϕ,ψ dξ. The above shift of the order of integration is justified by Fubini s theorem and the fact that for any h L 2 R 0 = R ĝξ) e δ2 λ1 iξ) d E λ h 2 2 dξ ĝξ) dξ e δ2λ d E λ h 2 2 ĝ 1 h To go further, we use that e δ2 1 iξ)l is an integral operator with kernel p z x, y), z = δ 2 1 iξ), and p z c to obtain for ϕ,ψ L 1 L 2 g δ 2 L ) e δ2l ϕ,ψ = 1 2π = R ĝξ) p δ 2 1 iξ) x, y)φx)ψy)dμx)dμy) )dξ [ 1 ĝξ)p 2π δ 2 1 iξ) ]φx)ψy)dμx)dμy). x, y)dξ 3.5) R To justify the above shift of order of integration we again use Fubini s theorem and the fact that ĝξ) pδ 2 1 iξ) x, y) φx) ψy) dμx)dμy)dξ R c ĝ 1 ϕ 1 ψ 1 <. This also implies gδ 2 L)e δ2l ϕ,ψ c ĝ 1 ϕ 1 ψ 1 for all ϕ,ψ L 1 L 2. Therefore, gδ 2 L)e δ2l is a kernel operator and by 3.5) g δ 2 L ) e δ2l x, y) = 1 ĝu)p 2π δ 2 1 iu)x, y)du. 3.6) R

24 From this and 1.5) we infer g δ 2 L ) e δ2l x, y) c Bx,δ) By,δ) ) 1/2 ĝu) { } exp cρ2 x, y) R δ u 2 du. ) 3.7) Assume ρx,y)/δ 1. Clearly, sup x 0 x β e x = β e )β for β>0. Using this with β = σ/2 we obtain { } exp cρ2 x, y) δ u 2 ) { ) } exp 1 + ρ2 x, y) c Therefore, δ u 2 ) ) c 1 + ρ2 x, y) σ/2 1 + u 2 ) σ/2 c δ ρx,y) ) σ ) σ 1 + u. δ g δ 2 L ) e δ2l x, y) c1 + ρx,y) δ ) σ Bx,δ) By,δ) ) 1/2 = c R ) ĝu) σ 1 + u dudσ,δ x, y), R ĝu) 1 + u ) σ du which confirms 3.2). If ρx,y)/δ < 1, then by 3.7) g δ 2 L ) e δ2l x, y) c Bx,δ) By,δ) ) 1/2 ĝu) du R c ĝu) 1 + u ) σ dudσ,δ x, y). R This completes the proof of 3.2). We now take on 3.3). As gδ 2 L)e δ2l = gδ 2 L)e 1 2 δ2l e 1 2 δ2l, the kernels of these operators are related by g δ 2 L ) e δ2l x, y) = g δ 2 L ) e 1 2 δ2l x, u)e 1 2 δ2l u, y)dμu), which implies g δ 2 L ) e δ2l x, y) g δ 2 L ) e δ2 L x,y ) g δ 2 L ) e 1 2 δ2l x, u) pδ 2 /2 u, y) p δ /2 2 u, y ) dμu). We use 3.2) with δ replaced by δ/ 2 and gλ) by g2λ) to estimate the first term under the integral and 1.6) for the second term, taking into account that

25 exp{ cρ2 x,y) δ 2 } c σ 1 + ρx,y) δ ) σ. Thus we get g δ 2 L ) e δ2l x, y) g δ 2 L ) e δ2 L x,y ) ρy,y ) ) α c g D δ,σ x, u)d δ,σ u, y)dμu) δ ρy,y ) ) α c g D δ,σx, y). δ Here for the latter estimate we used 2.11) and that σ>2d. It remains to prove 3.4). By 1.8), i.e. p δ 2 iu x, y)dy 1, and 3.6) we get g δ 2 L ) e δ2l x, y)dy = 1 ĝu) p 2π δ 2 iu x, y)dμy)du R = 1 2π R ĝu)du = g0). Here the justification of the shift of order of integration is by straightforward application of Fubini s theorem. Some remarks are in order. Condition 3.1) is apparently a smoothness condition on g. By Cauchy-Schwartz it follows that ĝξ) 1 + ξ 2 ) σ/2 dξ c ĝξ) ξ 2 ) ) 1/2 σ +1 dξ R R = c g H σ +1 and hence 3.1) holds if g H σ +1 <. However, it will be more convenient to us to replace 3.1) by a condition in terms of derivatives of g that is easier to verify. From ξ k ĝξ) = i) kĝk) ξ) we get ξ k ĝξ) g k) L 1.Also, ĝξ) g L 1. Pick k σ>2d. Then using the above we obtain ) k ξ ĝξ) 2 k+1 ĝξ) + ξ k+2 ĝξ) ) 2 k+1 g L 1 + g k+2) L 1) that implies ) g := ĝξ) kdξ 1 + ξ R = ĝξ) 1 + ξ ) k+2 ) 2dξ 1 + ξ R Thus we arrive at the following c g L 1 + g k+2) L 1).

26 Remark 3.2 For the norm g from condition 3.1) wehave g c g H σ +1 and g c g L 1 + g k+2) L 1) if k σ>2d. Corollary 3.3 For any m N and σ>0there exists a constant c σ,m > 0 such that the kernel of the operator L m e δ2l,0<δ 1, satisfies L m e δ2l x, y) c σ,m δ 2m D δ,σ x, y) and 3.8) L m e δ2l x, y) L m e δ2 L x,y ) ρy,y cσ,m δ 2m ) ) α D δ,σx, y), 3.9) δ if ρy,y ) δ. Proof Set gλ) := λ m θλ)e λ for λ 0, where θ C R), supp θ [ 1, ), and θλ)= 1forλ 0. Since L 0, we can write L m e δ2l = 2 m δ 2m g δ 2 L) e δ2 L with δ := 2 1/2 δ and the corollary follows by Theorem 3.1 and 2.8). We next use Theorem 3.1 and Remark 3.2 to obtain some important kernel localization results. Our main interest is in operators of the form fδ L). Theorem 3.4 Let f C 2k+4 R + ), k>2d, supp f [0,R] for some R 1, and f 2ν+1) 0) = 0 for ν = 0,...,k+1. Then fδ L),0<δ 1, is an integral operator with kernel fδ L)x, y) satisfying fδ L)x, y) c k D δ,k x, y) and 3.10) fδ L)x, y) fδ L) x,y ) ρy,y ) ) α D δ,kx, y) if ρ y,y ) δ, 3.11) c k δ where c k = c k f ) = c k R 2k+d+4 f L + f 2k+4) L +max ν 2k+4 f ν) 0) ) with c k > 0 a constant depending only on k,d, and the constants in 1.5) 1.6), and c k = c k R α ; as before α>0 is the constant from 1.6). Furthermore, fδ L)x, y)dμy) = f0) x. 3.12) Proof We first observe that it suffices to only prove the theorem when R = 1, then in the general case it follows by rescaling. Indeed, assume that f satisfies the hypotheses of the theorem and set hλ) := frλ), λ R +. Then h verifies the assumptions with R = 1 and if the theorem holds for R = 1 we obtain, using 2.8), fδ L)x, y) = h δr 1 L ) x, y) c k h)d δ/r,k x, y) 2R) d c k h)d δ,k x, y) 3.13)

27 and similarly fδ L)x, y) fδ L) x,y ) 2R) d+α c k h) ρy,y ) δ ) α D δ,kx, y) if ρ y,y ) δ R. For R δ <ρy,y ) δ, the last estimate follows by 3.13). It remains to observe that c k h) = c k f L + R 2k+4 f 2k+4) L + max Rν f ν) 0) ) ν 2k+4 cr 2k+4 f L + f 2k+4) L + max ν 2k+4 f ν) 0) ) and hence the theorem holds in general. We now prove the theorem in the case when R = 1. Choose θ C R) so that θ is even, supp θ [ 1, 1], θλ) = 1forλ [ 1/2, 1/2], and 0 θ 1. Denote P k λ) := k+2 f 2j) 0) j=0 2j)! λ 2j and let f 1 λ), g 0 λ), and g 1 λ) be defined for λ R + from fλ)= θλ)p k λ) + f 1 λ), θλ)p k λ) = g 0 λ 2 ) e λ2, f 1 λ) = g 1 λ 2 ) e λ2. Thus g 0 λ) = P k λ )θ λ )e λ for λ R +, and we use this to define g 0 λ) for λ<0. Clearly, g 0 C R), supp g 0 [ 1, 1] and g 0 L 1 + k+2) g 0 L 1 ck) sup ν 2k+4 f ν) 0). Therefore, by Theorem 3.1 the kernel of the operator θδ L)P k δ L) satisfies the desired inequalities 3.10) 3.11) with R = 1. On the other hand, g 1 λ) = f 1 λ )e λ for λ R + and we use this to define g 1 λ) for λ<0. Observe that f 1 δ L) = g 1 δ 2 L)e δ2l and supp g 1 [ 1, 1]. Furthermore, f 1 C 2k+4 R + ), f ν) 1 0) = 0, ν = 0,...,2k + 4, and j) f 1 L f j) L + c max f ν) 0), 0 j 2k ) ν 2k+4 We next show that g 1 C k+2 R) and estimate the derivatives of g 1.Wehavefor 1 m k + 2 and λ>0 g m) 1 λ) = m ν=0 ) ) m d ν e λ [ f1 λ) ] ν dλ and a little calculus shows that for ν 1 and λ>0 ) d ν [ f1 λ) ] = dλ ν j=1 c j λ ν+j/2 f j) 1 λ), where c j ν!.

28 On the other hand, by Taylor s theorem f j) 1 λ) λ 2m j)/2 f 2m) 1 L and hence ) d ν [ f1 λ ) ] c λ m ν f 2m) dλ 1, 1 ν m. Exactly in the same way we obtain the same estimate for λ<0. Denote briefly hλ) := f 1 λ ). Observe that since f 1 C 2k+4 R + ) we have h k+2) λ) = o1) as λ 0. This and the above inequalities yield h ν) 0) = 0, ν = 0,...,k + 2, and hence h C k+2 R), which implies g 1 C k+2 R). From the above we also obtain m) g 1 λ) m c e λ λ m ν 2m) f 1 L cm + 1) 2m) f 1 L, λ R. ν=0 This in turn with m = k + 2) implies g k+2) 1 L 1 ck + 3) f 2k+4) 1 L and, evidently, g 1 L 1 e f 1 L. We now apply Theorem 3.1 to conclude that f 1 δ L) is an integral operator with kernel f 1 δ L)x, y) satisfying 3.10) 3.11), where, in view of Remark 3.2 and 3.14), the constants c k, c k are of the claimed form. Putting the above together we conclude that fδ L) is an integral operator with kernel fδ L)x, y) satisfying 3.10) 3.11) with R = 1. Identity 3.12) follows by 3.4). Corollary 3.5 Let f : R + C be as in the hypothesis of Theorem 3.4. Then for any m N and 0 <δ 1 the operator L m fδ L) is an integral operator with kernel L m fδ L)x, y) such that L m fδ L)x, y) c k,m δ 2m D δ,k x, y) and 3.15) L m fδ L)x, y) L m fδ L) x,y ) ρy,y c k,m ) ) α δ 2m D δ,kx, y) 3.16) δ whenever ρy,y ) δ. Here the constants c k,m, c k,m are as the constants c k, c k in Theorem 3.4 with R 2k+d+4 replaced by R 2k+d+4+2m and c k depending on m as well. Proof Let hλ) := λ 2m fλ). Then hδ L) = δ 2m L m fδ L) and observe that h 2ν+1) 0) = 0forν = 0,...,k + 1. Consequently, the corollary follows by Theorem 3.4 applied to h. Corollary 3.6 Let f : R + C be as in the hypothesis of Theorem 3.4. Then there exists a constant c>0 such that for any 0 <δ 1 fδ L)φ q cδ 1/p 1/q φ p, φ L p, 1 p q, and fδ L)φx) fδ L)φy) ) ρx,y) α c φ, x,y, φ L. δ This corollary is an immediate consequence of Theorem 3.4 and Proposition 2.6.

29 3.2 Non-smooth Functional Calculus We need to establish some properties of operators of the form f L) and their kernels in the case of non-smooth compactly supported functions f. These are kernel operators with not necessarily well localized kernels. Theorem 3.7 Let f be a bounded measurable function on R + with supp f [0,τ] for some τ 1. Then f L) is an integral operator with kernel f L)x, y) satisfying ) f L x, y) c f, x,y, 3.17) Bx,τ 1 ) By,τ 1 ) and for x,y,y f L)x, y) f L) x,y ) c[τρy,y )] α f B x,τ 1 ) B y,τ 1 ) if ρ y,y ) τ ) Furthermore, if 1 p 2 q, f L) p q cτ d1/p 1/q) f, 3.19) f L),x) 2 2 = f 2 L)x, x) c B x,τ 1) 1 f 2, and 3.20) f 2 L) 1 = sup f 2 L)x, x). x 3.21) Above the constants depend only on d and the constants in 1.5) and 1.6); the constant in 3.19) depends in addition on p,q. Proof Pick a function θ C R + ) so that supp θ [0, 2], θx) = 1forx [0, 1], and 0 θ 1. Then by Theorem 3.4 θ τ 1 L ) x, y) c σ D τ 1,σ x, y) for any σ> ) Choose σ>3d/2. We have f L) = = 0 0 f λ)de λ θ τ 1 λ ) f λ)θ τ 1 λ ) de λ = θ τ 1 L ) f L)θ τ 1 L ). 3.23) Now, 3.17) follows by Proposition 2.9, using the above, 3.22), and the fact that f L) 2 2 f.

30 From 3.22) 3.23) and Proposition 2.9 we also obtain for 1 p 2 q f L) p q θ τ 1 L ) p 2 f L) 2 2 θ τ 1 L ) 2 q c f τ d1/q 1/p), which confirms 3.19). For the proof of 3.18), we first observe that f L) = = 0 0 f λ)e τ 2 λ) 2 e τ 2λ de λ g λ)e τ 2λ de λ = g L)e τ 2L, where gu) := fu)e τ 2 u 2, g e f, and hence f L)x, y) f L) x,y ) = g L)x, u) [ e τ 2L u, y) e τ 2 L u, y )] dμu). We now use 3.17), applied to g L), and 1.6) to obtain f L)x, y) f L) x,y ) c τρ y,y )) α g 1 e τρu,y))2 Bx,τ 1 ) B u, τ 1) Bu,τ 1 ) By,τ 1 ) dμu) cτρy,y )) α f e τρu,y))2 Bx,τ 1 ) By,τ 1 ) Bu,τ 1 ) dμu). oreover, using 2.2) wehave e τρu,y))2 Bu,τ 1 ) dμu) 2 d ) de 1+τρu,y) τρu,y)) 2 By,τ 1 dμu) c<, ) where for the latter inequality we used 2.12). This completes the proof of 3.18). We now turn to the proof of 3.20). We have f L),y) 2 2 = f L)x, y) 2 dy = f L)x, y)f L)x, y)dμy)

31 = f L)x, y)f L)y, x)dμy) = f 2 L)x, x) c B x,τ 1) 1 f 2, which proves 3.20). Here for the latter estimate we used 3.17). Finally, using the above we have f 2 L)x, y) = f L)x, u)f L)y, u)dμu) f L)x, u) ) 1/2 2 dμu) = f 2 L)x, x) ) 1/2 f 2 L)y, y) ) 1/2 f L)y, u) 2 dμu) ) 1/2 and hence f 2 L) 1 = sup x,y f 2 L)x, y) =sup x f 2 L)x, x), which confirms 3.21). 3.3 Approximation of the Identity and Littlewood-Paley Decomposition We first give a convenient approximation of the identity in L p statement. Proposition 3.8 Let ϕ C R + ), supp ϕ [0,R], R>0, ϕ0) = 1, and ϕ 2ν+1) 0) = 0 for ν = 0, 1,... Then for any f L p,1 p, L := UCB) one has f = lim ϕδ L)f in L p. δ 0 Proof By Theorem 3.4 it follows that ϕδ L) is an integral operator with kernel ϕδ L)x, y) satisfying for any k>2d ϕδ L)x, y) c k D δ,k x, y) c Bx,δ) δ 1 ρx,y) ) k+d/2, 3.24) where for the last inequality we used 2.2). Now, just as in the proof of 2.12) we obtain for k>3d/2 and r>0 ϕδ L)x, y) dμy) cδ/r) k 3d/2 0 as δ 0. \Bx,r) Indeed, suppose 2 l 1 δ r<2 l δ and denote E j := Bx,2 j δ) \ Bx,2 j 1 δ). Then using 3.24) and 2.1) we get ϕδ L)x, y) dμy) \Bx,r) c Bx,δ) 1 j l E j 1 + δ 1 ρx,y) ) k+d/2 dμy)

32 c Bx,δ) 1 j l Bx,2 j δ) j ) k d/2 c2 lk 3d/2) cδ/r) k 3d/2. On the other hand, from 3.12) and ϕ0) = 1wehave ϕδ L)x, y)dμy) = 1. Using the above and the fact that the vector lattice set of all boundedly supported uniformly continuous functions on is dense in L p by the Stone-Daniell theorem) one proves as usual the claimed convergence. We next give precise meaning to what we call Littlewood-Paley decomposition of L p -functions in this article. Corollary 3.9 Let ϕ 0,ϕ C R + ), supp ϕ 0 [0,b] and supp ϕ [b 1,b] for some b>1, ϕ0) = 1, ϕ 2ν+1) 0) = 0 for ν 0, and ϕ 0 λ) + j 1 ϕb j λ) = 1 for λ R +. Then for any f L p,1 p, L := UCB) f = ϕ 0 L) + j 1 ϕ b j L ) f in L p. 3.25) Proof Let θλ) := ϕ 0 λ) + ϕb 1 λ) and observe that j k=0 ϕ kλ) = θb j λ) for j 1. Then the result follows by Proposition Spectral Spaces We adhere to the setting of this article, described in the introduction. As before E λ, λ 0, is the spectral resolution associated with the self-adjoint positive operator L on L 2 := L 2, μ). As elsewhere we shall be dealing with operators of the form f L). We denote by F λ, λ 0, the spectral resolution associated with L, that is, F λ = E λ 2. Then f L) = 0 fλ)df λ and the spectral projectors are defined by E λ = 1 [0,λ] L) := 0 1 [0,λ] u)de u and F λ = 1 [0,λ] L) := 0 1 [0,λ] u)df u = 0 1 [0,λ] u)de u. 3.26) We next list some properties of F λ which follow readily from Theorem 3.7: The operator F λ is a kernel operator whose kernel F λ x, y) is a real symmetric nonnegative function on. Also, F λ x, y) c B x,λ 1) 1/2 B y,λ 1) 1/2 3.27) and F λ x, y) is in Lip α for some α>0, see 3.18). The mapping property of F λ on L p spaces is given by F λ f q cλ d1/p 1/q) f p, 1 p 2 q. 3.28)